Variational Representation and Convexity of the Matrix Inverse

Let $k\ge1$ be a natural number. All matrices below are real $k\times k$ matrices, combined entrywise by the matrix sum and scalar multiple; $x\cdot y$ denotes the dot product on the Euclidean space $\mathbb{R}^{k}$, and $Mx$ denotes the matrix-vector product.

1. (Variational representation) Let $M$ be a symmetric positive definite matrix, with the inverse $M^{-1}$ provided by Invertibility of Symmetric Positive Definite Matrices. Then for all $x,y\in\mathbb{R}^{k}$,

with equality if and only if $y=M^{-1}x$.

2. (Convexity of the inverse) Let $A$ and $B$ be symmetric positive definite matrices and let $\lambda$ be a real number with $0\le\lambda\le1$. Then $\lambda A+(1-\lambda)B$ is symmetric positive definite, and in the semidefinite order